Mauricio Pochettino's side had gone four games without a victory before Wednesday's tie, including home defeats against Liverpool and West Brom.
And Spurs looked in danger of suffering more White Hart Lane pain when 19-year-old Jorge Grant put Championship leaders Forest ahead with his first senior goal on 61 minutes.
However, Spurs responded with strikes from Ryan Mason, Roberto Soldado and Harry Kane to complete a 3-1 comeback victory before being drawn at home to Brighton in round four.
"We knew it was going to be tough, Nottingham Forest are top of the Championship and they were obviously going to go for it in a cup game, so we knew it would be difficult," Townsend told Sky Sports.
"We went a goal behind but we showed great spirit and the quality showed at the end of the game.
"It's a very important win. The performances have been okay, but the results have not been there, so it was good to get a morale-boosting win with the big game coming up on Saturday.
"Of course, the manager made a lot of changes, but the lads that played showed that they are good enough to play if selected on Saturday, so I think the manager has got a few selection problems - although I'm sure he'll want that."
It was certainly a special night for 23-year-old midfielder Mason, who has had spells out on loan with a number of clubs before getting his chance to shine for Tottenham.
"I've been waiting for that for a very long time and it was obviously nice to come on and get the goal with the way the game was going," he commented.
"I've always been confident in my ability, I've had to wait a lot longer than a lot of other players, but I can't moan, I've come on tonight and I thought I made a difference."